A261156 Expansion of chi(q) * chi(-q^9) / (chi(-q) * chi(q^9)) in powers of q where chi() is a Ramanujan theta function.

1, 2, 2, 4, 6, 8, 12, 16, 22, 28, 36, 48, 60, 76, 96, 120, 150, 184, 228, 280, 340, 416, 504, 608, 732, 878, 1052, 1252, 1488, 1768, 2088, 2464, 2902, 3408, 3996, 4672, 5460, 6364, 7400, 8600, 9972, 11544, 13344, 15400, 17752, 20424, 23472, 26944, 30876, 35346

Offset: 0

Author

Michael Somos, Aug 10 2015

Keywords

• nonn

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + 6*x^4 + 8*x^5 + 12*x^6 + 16*x^7 + 22*x^8 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A080054, A123629, A186924, A187100, A212484, A233693, A260215.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^2] QPochhammer[ -q, q] QPochhammer[ q^9, q^18] QPochhammer[ q^9, -q^9], {q, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^9 + A)^2 * eta(x^36 + A) / (eta(x + A)^2 * eta(x^4 + A) * eta(x^18 + A)^3), n))};

Formula

Expansion of eta(q^2)^3 * eta(q^9)^2 * eta(q^36) / (eta(q)^2 * eta(q^4) * eta(q^18)^3) in powers of q.

G.f. A(x) = B(x) / B(x^9) where B(x) is the g.f. of A080054.

Euler transform of period 36 sequence [ 2, -1, 2, 0, 2, -1, 2, 0, 0, -1, 2, 0, 2, -1, 2, 0, 2, 0, 2, 0, 2, -1, 2, 0, 2, -1, 0, 0, 2, -1, 2, 0, 2, -1, 2, 0, ...].

a(n) = 2 * A233693(n) unless n=0. a(2*n) = 2 * A123629(n) = 2 * A212484(n) unless n=0.

a(3*n) = A186924(n). a(3*n) = 4 * A187100(n) unless n=0.

a(n) = (-1)^n * A260215(n). - Michael Somos, Aug 14 2015

a(n) ~ exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017